Dualization Problem over the Product of Chains: Asymptotic Estimates for the Number of Solutions

A key intractable problem in logical data analysis, namely, dualization over the product of partial orders, is considered. The important special case where each order is a chain is studied. If the cardinality of each chain is equal to two, then the considered problem is to construct a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form, which is equivalent to the enumeration of irreducible coverings of a Boolean matrix. The asymptotics of the typical number of irreducible coverings is known in the case where the number of rows in the Boolean matrix has a lower order of growth than the number of columns. In this paper, a similar result is obtained for dualization over the product of chains when the cardinality of each chain is higher than two. Deriving such asymptotic estimates is a technically complicated task, and they are required, in particular, for proving the existence of asymptotically optimal algorithms for the problem of monotone dualization and its generalizations.     

Representations of torsion-free arithmetic matroids

We study the representability problem for torsion-free arithmetic matroids. After introducing a “strong gcd property” and a new operation called “reduction”, we describe and implement an algorithm to compute all essential representations, up to equivalence. As a consequence, we obtain an upper bound to the number of equivalence classes of representations. In order to rule out equivalent representations, we describe an efficient way to compute a normal form of integer matrices, up to left-multiplication by invertible matrices and change of sign of the columns (we call it the “signed Hermite normal form”). Finally, as an application of our algorithms, we disprove two conjectures about the poset of layers and the independence poset of a toric arrangement.     

Column Generation Algorithms for Nonlinear Optimization,I: Convergence Analysis

Column generation is an increasingly popular basic tool for the solution of large-scale mathematical programming problems. As problems being solved grow bigger, column generation may however become less efficient in its present form, where columns typically are not optimizing, and finding an optimal solution instead entails finding an optimal convex combination of a huge number of them. We present a class of column generation algorithms in which the columns defining the restricted master problem may be chosen to be optimizing in the limit, thereby reducing the total number of columns needed. This first article is devoted to the convergence properties of the algorithm class, and includes global (asymptotic) convergence results for differentiable minimization, finite convergence results with respect to the optimal face and the optimal solution, and extensions of these results to variational inequality problems. An illustration of its possibilities is made on a nonlinear network flow model, contrasting its convergence characteristics to that of the restricted simplicial decomposition (RSD) algorithm.     

High‐performance direct algorithms for computing the sign function of triangular matrices

Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache‐efficient algorithms that are based on Higham's stabilized specialization of Parlett's substitution algorithm for computing the sign of a triangular matrix. We show that the new recursive algorithms are asymptotically optimal in terms of the number of cache misses that they generate. One algorithm that we present performs more arithmetic than the nonrecursive version, but this allows it to benefit from calling highly optimized matrix multiplication routines; the other performs the same number of operations as the nonrecursive version, suing custom computational kernels instead. We present implementations of both, as well as a cache‐efficient implementation of a block version of Parlett's algorithm. Our experiments demonstrate that the blocked and recursive versions are much faster than the previous algorithms and that the inertia strongly influences their relative performance, as predicted by our analysis.     

Genetic algorithms applied to the solution of hybrid optimal control problems in astrodynamics

Many space mission planning problems may be formulated as hybrid optimal control problems, i.e. problems that include both continuous-valued variables and categorical (binary) variables. There may be thousands to millions of possible solutions; a current practice is to pre-prune the categorical state space to limit the number of possible missions to a number that may be evaluated via total enumeration. Of course this risks pruning away the optimal solution. The method developed here avoids the need for pre-pruning by incorporating a new solution approach using nested genetic algorithms; an outer-loop genetic algorithm that optimizes the categorical variable sequence and an inner-loop genetic algorithm that can use either a shape-based approximation or a Lambert problem solver to quickly locate near-optimal solutions and return the cost to the outer-loop genetic algorithm. This solution technique is tested on three asteroid tour missions of increasing complexity and is shown to yield near-optimal, and possibly optimal, missions in many fewer evaluations than total enumeration would require.     

Minimising the number of gap-zeros in binary matrices

We study a problem of minimising the total number of zeros in the gaps between blocks of consecutive ones in the columns of a binary matrix by permuting its rows. The problem is referred to as the Consecutive Ones Matrix Augmentation Problem, and is known to be NP-hard. An analysis of the structure of an optimal solution allows us to focus on a restricted solution space, and to use an implicit representation for searching the space. We develop an exact solution algorithm, which is linear-time in the number of rows if the number of columns is constant, and two constructive heuristics to tackle instances with an arbitrary number of columns. The heuristics use a novel solution representation based upon row sequencing. In our computational study, all heuristic solutions are either optimal or close to an optimum. One of the heuristics is particularly effective, especially for problems with a large number of rows.     

A mathematical programming-based scheduling framework for multitasking environments

A new paradigm along with a mixed (binary) integer-linear programming model is developed for scheduling tasks in multitasking environments, for which the number of completed tasks is not a good measure. One special case falls into the realm of deteriorating jobs. Polynomial time optimal solution algorithms are presented for this and one other special case. As the complexity of the original problem is believed to be strongly NP-hard, an efficient solution algorithm, based on tabu search, is developed to solve the problem. Small, medium, and large size problems are solved, and the solution obtained from the algorithm is compared with that of the optimal solution or the upper bound found from using the Lagrangian relaxation. Where it was measurable, the search algorithm gave quantifiably good quality solutions, and in all cases it had a much better time efficiency than the branch-and-bound enumeration method. A detailed statistical experiment, based on the split-plot design, is developed to identify the characteristics of the tabu search algorithm, thus guaranteeing a solution that is significantly better in quality. A conjecturing technique is introduced for problems with very large planning horizons. This technique had remarkable time efficiency with no apparent loss of quality.     

Modifications of the standard vector Monte Carlo estimate for characteristics analysis of scattered polarized radiation

There are two versions of weighted vector algorithms for the statistical modeling of polarized radiative transfer: a “standard” one, which is convenient for parametric analysis of results, and an “adaptive” one, which ensures finite variances of estimates. The application of the adaptive algorithm is complicated by the necessity of modeling the previously unknown transition density. An optimal version of the elimination algorithm used in this case is presented in this paper. A new combined algorithm with a finite variance and an algorithm with a mixed transition density are constructed. The comparative efficiency of the latter is numerically studied as applied to radiative transfer with a molecular scattering matrix.     

Efficient algorithms for the recognition of topologically conjugate gradient-like diffeomorhisms

It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NPcomplete). We give polynomial-time algorithms for recognition of the corresponding graphs for two gradient-like systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, point-source potential-field models in efforts to determine the nature of the quiet-Sun magnetic field.     

带时间窗取送货问题的混合算法

为解决带时间窗的取送货问题,建立了集合划分模型,设计列生成算法与启发式规则相结合的CGA混合算法进行求解。首先,放松约束构建主问题及受限主问题,运用单纯形法与分支定界进行求解;其次,建立时空网络以构建子问题,基于修正的Dijkstra's算法,设计包含算法A、B1、B2的求解算法;最后,通过启发式算法解决节点重复覆盖问题。为验证算法有效性,进一步构建了OPT近似最优解算法;并基于CGA提出三种求解策略C1、C2、C3,做单因素方差分析,采用算例分析算法的性能。实验结果表明,对于客户点数量小于30的小规模算例,CGA与OPT所得结果相近,但CGA求解效率更显著;针对客户点数量为600的大规模算例,CGA至多在20分钟内求得结果,可见本文算法的精度和效率较高。而针对不同类型及规模的客户点的单因素方差分析结果显示,C1、C2、C3在“平均行驶距离成本”、“平均车辆数”、“平均求解时间”三个维度上差异性显著,经营者可根据实际需求进行策略选择。     

Block rearranging elements within matrix columns to minimize the variability of the row sums

Several problems in operations research, such as the assembly line crew scheduling problem and the k-partitioning problem can be cast as the problem of finding the intra-column rearrangement (permutation) of a matrix such that the row sums show minimum variability. A necessary condition for optimality of the rearranged matrix is that for every block containing one or more columns it must hold that its row sums are oppositely ordered to the row sums of the remaining columns. We propose the block rearrangement algorithm with variance equalization (BRAVE) as a suitable method to achieve this situation. It uses a carefully motivated heuristic—based on an idea of variance equalization—to find optimal blocks of columns and rearranges them. When applied to the number partitioning problem, we show that BRAVE outperforms the well-known greedy algorithm and the Karmarkar–Karp differencing algorithm.     

An approach for solving a class of transportation scheduling problems

An algorithm is developed for solving a class of transportation scheduling problems. It applies for a variety of problems such as: the Combining Truck Trip problem, the Delivery problem, the School Bus problem, the Assignment of Buses to Schedules, and the Travelling Salesman problem. The objective functions of the above problems differ from each other. Yet, by using the “savings method” proposed by Clarke and Wright, and extended by Gaskell, we are able to define each one of the above problems as a series of assignment problems. The cost matrix entries of each one of the assignment problems are a function of the constraints of the particular routing or scheduling problem. The solution to the assignment problem determines an upper bound of the optimal solution to the original problem. By combining the above procedure with a Branch and Bound procedure, it is possible to obtain the optimal solution in a finite number of steps. In some cases the Branch and Bound process can be eliminated due to the nature of the problem and in those cases the algorithm is efficient.     

Scalable and fault tolerant orthogonalization based on randomized distributed data aggregation

The construction of distributed algorithms for matrix computations built on top of distributed data aggregation algorithms with randomized communication schedules is investigated. For this purpose, a new aggregation algorithm for summing or averaging distributed values, the push-flow algorithm, is developed, which achieves superior resilience properties with respect to failures compared to existing aggregation methods. It is illustrated that on a hypercube topology it asymptotically requires the same number of iterations as the optimal all-to-all reduction operation and that it scales well with the number of nodes. Orthogonalization is studied as a prototypical matrix computation task. A new fault tolerant distributed orthogonalization method rdmGS, which can produce accurate results even in the presence of node failures, is built on top of distributed data aggregation algorithms.     

A combinatorial optimization algorithm for solving the branchwidth problem

In this paper, we consider the problem of computing an optimal branch decomposition of a graph. Branch decompositions and branchwidth were introduced by Robertson and Seymour in their series of papers that proved the Graph Minors Theorem. Branch decompositions have proven to be useful in solving many NP-hard problems, such as the traveling salesman, independent set, and ring routing problems, by means of combinatorial algorithms that operate on branch decompositions. We develop an implicit enumeration algorithm for the optimal branch decomposition problem and examine its performance on a set of classical graph instances.     

Locally restricted compositions over a finite group

One may generalize integer compositions by replacing the positive integers with a different additive semigroup, giving the broader concept of a “composition over a semigroup”. Here we focus on semigroups which are finite groups and achieve asymptotic enumeration of compositions over a finite group which satisfy a local restriction. These compositions are associated to walks on a voltage graph whose structure is exploited to simplify asymptotic expressions. Specifically, we show that under mild conditions the number of locally restricted compositions of a group element is asymptotically independent of the particular group element. We apply this result to subword pattern avoidance and other examples such as generalized Carlitz compositions.     

Asymptotically Optimal Algorithms for Job Shop Scheduling and Packet Routing

We propose asymptotically optimal algorithms for the job shop scheduling and packet routing problems. We propose a fluid relaxation for the job shop scheduling problem in which we replace discrete jobs with the flow of a continuous fluid. We compute an optimal solution of the fluid relaxation in closed form, obtain a lower bound C_max to the job shop scheduling problem, and construct a feasible schedule from the fluid relaxation with objective value at most where the constant in the O( · ) notation is independent of the number of jobs, but it depends on the processing time of the jobs, thus producing an asymptotically optimal schedule as the total number of jobs tends to infinity. If the initially present jobs increase proportionally, then our algorithm produces a schedule with value at most C_max + O(1). For the packet routing problem with fixed paths the previous algorithm applies directly. For the general packet routing problem we propose a linear programming relaxation that provides a lower bound C_max and an asymptotically optimal algorithm that uses the optimal solution of the relaxation with objective value at most Unlike asymptotically optimal algorithms that rely on probabilistic assumptions, our proposed algorithms make no probabilistic assumptions and they are asymptotically optimal for all instances with a large number of jobs (packets). In computational experiments our algorithms produce schedules which are within 1% of optimality even for moderately sized problems.     

Fast scaling algorithms for M-convex function minimization with application to the resource allocation problem

M-convex functions, introduced by Murota (Adv. Math. 124 (1996) 272; Math. Prog. 83 (1998) 313), enjoy various desirable properties as “discrete convex functions.” In this paper, we propose two new polynomial-time scaling algorithms for the minimization of an M-convex function. Both algorithms apply a scaling technique to a greedy algorithm for M-convex function minimization, and run as fast as the previous minimization algorithms. We also specialize our scaling algorithms for the resource allocation problem which is a special case of M-convex function minimization.     

Parallel defect control

How can small-scale parallelism best be exploited in the solution of nonstiff initial value problems? It is generally accepted that only modest gains inefficiency are possible, and it is often the case that “fast” parallel algorithms have quite crude error control and stepsize selection components. In this paper we consider the possibility of using parallelism to improvereliability andfunctionality rather than efficiency. We present an algorithm that can be used with any explicit Runge-Kutta formula. The basic idea is to take several smaller substeps in parallel with the main step. The substeps provide an interpolation facility that is essentially free, and the error control strategy can then be based on a defect (residual) sample. If the number of processors exceeds (p ? 1)/2, wherep is the order of the Runge-Kutta formula, then the interpolant and the error control scheme satisfy very strong reliability conditions. Further, for a given orderp, the asymptotically optimal values for the substep lengths are independent of the problem and formula and hence can be computed a priori. Theoretical comparisons between the parallel algorithm and optimal sequential algorithms at various orders are given. We also report on numerical tests of the reliability and efficiency of the new algorithm, and give some parallel timing statistics from a 4-processor machine.     

Performance evaluation of fault-tolerant routing algorithms: an optimization problem

In this paper, we present a new theoretical approach for studying the behaviour and the performance of shortest paths fault-tolerant distributed algorithms of a certain class. The behaviour of each processor is modeled by means of a stochastic matrix. We show that achieving the optimal behaviour of Nprocessors is equivalent to solvingan optimization problem of a function of 2N variables under constraints; this function is neither convex nor concave. Solutions for which such a type of algorithms has an optimal behaviour are derived. Using that result, we build a fuzzy set of solutions which provides a global overview (a sort of “relief”): each solution of the fuzzy set has value ? ranging between 0 and 1, which may be regarded as its“bench-mark” so (1 -?) points out the proximity of any solution from the optimal solution